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The Polynomial Hirsch Conjecture: The Crux of the Matter.

Suppose that (*) For every , and every and , there is which contains .

The basic question is: How large can t be???

Let’s call the answer f(n).

Remark: If you restrict your attention to sets in these families containing an element m and delete m from all of them, you get another example of such families of sets, possibly with smaller value of t. (Those families which do not include any set containing m will vanish.)

Theorem:.

Proof: Consider the largest s so that the union of all sets in is at most n/2. Clearly, .
Consider the largest r so that the union of all sets in is at most n/2. Clearly, .

Now, by the definition of s and r, there is an element m shared by a set in the first s+1 families and a set in the last r+1 families. Therefore (by (*)), when we restrict our attention to the sets containing ‘m’ the families all survive. We get that . Q.E.D.

Remarks:

1) The recurrence relation gives . The major task is to prove a polynomial upper bound for or to find examples showing that is superpolynomial in .

2) The abstract setting is taken from the paper Diameter of Polyhedra: The Limits of Abstraction by Freidrich Eisenbrand, Nicolai Hahnle, Sasha Razborov, and Thomas Rothvoss. We can consider families of d-subsets of {1,2,…, n}, and denote the maximum cardinality t by . The argument above gives the relation , which implies .

3) (and thus also ) are upper bounds for the diameter of graphs of d-polytopes with n facets.

5 thoughts on “The Polynomial Hirsch Conjecture: The Crux of the Matter.”

Is there any possibility that all mathematical formulas, concepts, proofs can only get solved ate some single point in the history of mankind: be it past, present or future? Would it require a cross germination of genes in the totality of humanity to combine into a new structure to fully make us understand what is happening in all branches of mathematics???

Would it have to be a ‘physical’ process: a genetic freak in a single human being to join all the dots together. Could this super special person be ‘you’???